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A Clifford algebra gauge invariant Lagrangian for
gravity. Part1: higher dimensions and reduction to
four-dimensional space-time
Jean Pierre Pansart
To cite this version:
Jean Pierre Pansart. A Clifford algebra gauge invariant Lagrangian for gravity. Part1: higher di-mensions and reduction to four-dimensional space-time. [Research Report] Commissariat à l’Energie Atomique, CEN Saclay, Irfu/SPP. 2016. �hal-01261519�
A Clifford algebra gauge invariant Lagrangian for gravity.
Part1: higher dimensions and reduction to four-dimensional
space-time.
J. P. Pansart 1
January 2016
Introduction.
Gauge fields have the nature of connexions which distinguishes them from matter fields, which in the present note will be spinor fields. A gauge transformation is a change of local reference frame, and the gauge invariance simply says that the description of physical
phenomena does not depend on the local reference frame used. Gauge invariance does not say anything on the nature of the space-time ! , but invariance with respect to gauge
transformations helps to build Lagrangians.
The main motivation of these notes is that the gravitational field should have the same nature as gauge fields. Like in Kaluza-Klein theories, the space-time will be supposed to be an ! dimensional manifold : ! locally of the form : ! , where ! is a compact space of small radius of curvature, invariant under the action of a group ! . Usually, in this type of theories, the Lagrangian driving the dynamics of ! is the Einstein-Hilbert one. We shall have a different point of view and shall consider the gauge fields associated to the elements of a graded Lie algebra built from the Clifford algebra represented by the Dirac matrices of the ! dimensional pseudo-euclidean space. This will be explained in chapter 3, which can be read first. The Lagrangian will be the usual quadratic Lagrangian of gauge fields. After reduction, and at the 4 dimensional space-time level, the gravity field Lagrangian will contain four terms : the Einstein-Hilbert Lagrangian given by the space-time scalar curvature, a quadratic term similar to the gauge field Lagrangian, a cosmological constant term and a torsion one. Torsion is naturally introduced and can propagate.
The goal of this note is to show that, after integration on ! , that is to say at the
macroscopic level, one obtains the classical Lagrangian of a spinor field with gravitational and gauge fields, and the usual quadratic Lagrangian of gauge fields. Only the gravitational field Lagrangian will be modified as described above. We show, in a separate note, that this modified Lagrangian gives the same dynamics as the Einstein-Hilbert one in weak
gravitational field situations. Differences may appear only in the case strong gravitational fields, but we have not yet studied these consequences.
The notations used in this note are presented in appendix B which recalls also some basic geometrical equations. Chapters 1 and 2 study respectively the consequences of the invariance under the action of the group ! , and the transformations of the spinors in
V n+ m V =Vn+m Vn⊗Vm Vm G V n+m Vm G
Retired from : Commissariat à l’Energie Atomique, CEN Saclay, DSM/ IRFU/SPP
1
! . As said above, chapter 3 presents the construction of the basic Lagrangian. Chapter 4 studies the reduction of the Dirac equation in ! in order to obtain the « macroscopic » Dirac equation with gauge fields and gravitational field in ! . Chapter 5 calculates the different components of the curvature tensor and show how the gauge field Lagrangian of chapter 3 is reduced. At last, chapter 6 look at the mass term of the Dirac equation. A few more hypotheses will be made along the the following chapters. Appendix A shows how to build a possible set of Dirac matrices in ! .
1. Hypotheses and first consequences.
1.1 Definitions and hypotheses.
As in Kaluza-Klein type theories , we shall assume that the space-time is an ! dimensional manifold ! , locally of the form : ! .
The space-time coordinates of a point ! are labelled with Greek letters ! : ! , ! . If it is necessary to distinguish the coordinates, the letters : ! are used if : ! , and : ! if : ! . Likewise, when tensor
components are expressed with respect to local orthonormal frames, we use Latin letters ! , and the indices : ! if : ! , and : ! if : ! . The symbols used in this note are defined in Appendix B, which provides also a very brief summary of the basic geometrical equations and definitions.
As in Kaluza-Klein theories, ! is assumed to be invariant under the action of a
transformation group ! whose parameters are called ! : ! . The group has no action on ! : ! .
We set : ! (1.1) therefore, from the hypothesis : ! (1.2) For an infinitésimal transformation : ! (1.3) ! , the transformation of the orthonormal local frame basis vectors is given by: ! (1.4) This formula is valid for any vector field, not only for the ! .
! can be considered as an hyper-surface embedded in ! , and one can chose orthonormal local frames such that ! , ! , are tangent to this hyper-surface, by setting : ! (1.5) With the condition (1.2) we then have : ! , which shows that the vectors ! remain tangent to ! in a transformation (1.3).
V =Vn+m Vn+m Vn V =Vn+m n+m V =Vn+m Vn⊗Vm x α ,β ,γ ...
{ }
xα 0≤α,β,γ, ...<n+m µ, ν, η, ρ 0≤α <n ϕ, τ, χ, ψ n≤α <n+m a, b, c, ... i, j, k, l, m 0≤a<n r, s, t, u n≤a<n+m Vm G{ }
ax x 'ϕ= fϕ(a, x) Vn x 'µ= fµ(a, x)=xµ Xx α = ∂fα(a, x) ∂ax |ax=0 Xx µ = 0 x 'α=xα +ηxXx α ηx ≪ 1 h 'a α(x ')=h a α(x ')− ηx Xx, ha ⎡⎣ ⎤⎦α ha !"!{ }
Vm V =Vn+m hr !"!{ }
n≤r<n+m hr µ =0 h'r µ(x ')=h r µ(x ') h r !"!{ }
VmWe write : ! , ! (1.6)
The differential forms describing the neighborhood of the point where the local frame is defined, are therefore :
! et : ! (1.7a) With : ! , one has : ! and : ! (1.7b) ! then : ! (1.7c) and the inverse relation : !
Likewise, with : ! , one has : ! and ! (1.7d) At last : ! , ! (1.7e) From (1.6) the space-time volume element is :
!
!
In Kaluza-Klein theories the gauge fields : ! come from the identification : ! (1.8) Therefore these gauge fields have not directly the nature of a connexion, however they transform as gauge fields under the action of the group ! , as it is now recalled. At first order, one has, using (1.4) :
! and with (1.8) :
!
! , which represents the internal action in ! of infinitesimal elements of ! , does not depend on ! , and if we assume that : ! , we obtain :
!
Then , if the transformation parameters depend only on the point in ! , ! : !
If : ! , and if : ! , the ! transforms like an element of the adjoint representation of the Lie algebra of the group ! .
1.2
First consequences of the invariance hypothesis.
In all what follows, the coefficients ! of the transformations (1.3) are constants, except if explicitly mentioned. We define : ! (1.9a) and : ! (1.9b) ha α = hi µ 0 hi ϕ hr ϕ ⎛ ⎝⎜ ⎞ ⎠⎟ hα a = hµ i 0 hµr hϕr ⎛ ⎝⎜ ⎞ ⎠⎟ ωi = hµi dxµ ωr = hµr dxµ +hϕr dxϕ ha α hαb =δa b hi µ hµj =δi j hr ϕ hϕs =δr s hi µ hµr =−hi ϕ hϕr hµr = −hµi hi ϕ hϕr hi ϕ =−h i µ h µrhr ϕ hαa ha β =δ αβ hµi hi ν =δ µ ν h ϕrhr τ =δ ϕτ dxµ =hi µωi dxϕ =hi ϕωi+ hr ϕωr dV =det(hαa) dxα0 ∧ ... ∧ dxαn+m−1 =(det(hµi) dxµ0 ∧ ... ∧ dxµn−1)∧ (det(h ϕr) dxϕ0 ∧ ... ∧ dxϕm−1)=dVn ∧ dVm Wνx (xν) hνr =hϕr Wνx(xν) Xx ϕ G h 'i ϕ (x)=hi ϕ (x)−⎡⎣ηxXx, hi⎤⎦ ϕ hi µ W 'µx Xx ϕ = hi µ WµxXx ϕ −ηx ( Xx τ∂ τ(hi ν WνyXy ϕ )− hi α∂ αXx ϕ )+Xx ϕ hi α∂ αηx Xx ϕ Vm G xµ
{ }
∂τhi µ = 0 hi µW ' µ x Xx ϕ =h i µW µxXx ϕ −ηx hi νW νy⎡⎣Xx, Xy⎤⎦−η x Xx τh i ν X y ϕ∂ τ(Wνy)+Xx ϕh i α∂ αηx Vn ηx =ηx (xν) W 'µx =Wµx +ηzWµyC. y z x −ηy Xy τ∂ τ(Wµx)+∂µηx ∂τ(Wµx)=0 ∂τhi µ = 0 Wµx G ηx Xx ! "! , ha !"! ⎡⎣ ⎤⎦=qx a . b hb !"! h 'a α(x)=h a α(x)−ηx qx a . b (x) hb α(x)! (1.9c) where the coefficients ! are the structure constants of ! .
Using the Jacobi identities, one can get a relation between the ! : !
then : ! (1.10) If ! was a rigid group of motion, one would have at first order :
!
and setting : ! , this gives the constraint :
! (1.11) Using the above hypotheses, we now look at the consequences for the coefficients ! . With (1.2) and (1.6) : ! (1.12a) which is a relation on ! only.
In the same way : ! (1.12b) ! (1.12c) which shows that, with (1.4), ! remains tangent to ! , in agreement with the hypotheses. At last, since ! does not depend on ! :
! (1.12d) If the torsion is not zero, an infinitesimal parallelogram does not close itself. From that property, and from (1.4), the action of a group of rigid motions would give the constraint : ! (1.13) 1.3
Commutators and connexion .
The relations (1.5) et (1.6) are constraints on the local orthonormal frame basis vectors. These vectors are related to the connexion coefficients through the structure equations: ! (1.14) In this section we examine the consequences of these constraints on the connexion coefficients. To do that, the connexion coefficients are written using the basis vector commutators. We define : ! (1.15a) then : ! (1.15b) (Although the same notation has been used for the commutator of the vectors ! , there will be no ambiguity thanks to the index letters). One has also :
! (1.15c) ! (1.15d) where : ! Xx, Xy ⎡⎣ ⎤⎦ =C. x y z Xz C. y x z G qx a . b Xy, X
[
x, ha]
⎡⎣ ⎤⎦ +⎡⎣ha, X⎡⎣ y, Xx⎤⎦⎤⎦ +⎡⎣Xx, h⎡⎣ a, Xy⎤⎦⎤⎦ =0 qyc. b qx a. c− qx c. b qya. c = C. y x z qz a. b+ Xx(qya. b )−Xy(qx a. b ) G h'a ! "! . h'b ! "! =ha !"! . hb !"! −ηx (qx a. cη cb+qx b. cη c a) qx a b =qx a . cη c b qx a b+qx b a=0 qx a . b qx r . s = (Xx τ ∂ τhr ϕ − h r τ ∂ τXx ϕ ) hϕs Vm qx i. j = Xx τ ∂ τhi µ h µj qx r . i= 0 h'r ! "! Vm Xx ϕ xµ qx i . r = (Xx τ ∂ τhi ϕ − hi τ ∂ τXx ϕ ) hϕr +Xx τ ∂ τhi µ hµr Xx(S. b c a )−S. b e a (x) qx c . e− S. e c a (x) qx b . e+ S. b c e (x) qx e . a = 0 dωa +ω. b a ∧ωb =− S . b c a ωb ∧ωc=Σa ha,hb ⎡⎣ ⎤⎦ =Cc.a bh c Cc.a b=Γ .ba c − Γ .a b c +2 S .a b c Xx Γcba− Γc ab =ηc dC.a b d −2 S c ab 2Γa bc =ηc dC.a bd −η bdC.c a d −η a dC.bc d +2 S a bc Sa bc=Sa bc−Sc a b− Sba c= −Sba cis the contorsion tensor.
The relation (1.15d) is true only for rigid local frames, and therefore valid for the local orthonormal frames used. Since in (1.15d) the torsion terms are clearly separated,
calculations can be performed in the case of zero torsion, and the terms corresponding to it added later. For rigid local frames one has : !
then (1.15c) is also : ! (1.15e) At last, with (1.15b) : ! (1.15f) In (1.15d) we set : ! (1.15g) which satisfy : ! (1.15h) As it was done for the: ! in (1.12) , one can now look at the commutation coefficients : !
and with (1.5) : ! (1.16a) Likewise : ! (1.16b) then : ! if ! (1.16c) ! is computed on ! only (1.16d) at last: ! is computed on ! only if : ! (1.16e) From (1.15e) , (1.16c) et (1.16a) , one immediately gets the following symmetry relations : ! if : ! and : ! (1.16f) moreover : ! is calculated on ! only (1.16g) and : ! is calculated on ! only , if : ! (1.16h) The other components are : ! (1.16i) ! (1.16j) ! if : ! (1.16k) Let us go back to the definition of the commutator of the local frame basis vectors: !
or equivalently : !
and set : ! (1.17) One obtains ! (1.18a) ! (1.18b) ! (1.18c) Γbac = −Γabc Γacb− Γbc a =ηc dC.a b d −2 S c ab C.a e e =− Γ .a e e +2 S .a e e 2Γa b c =ηc dC.a b d −η b dC.c a d −η a dC.b c d Γcba− Γc ab =ηc dC.a b d = Γ acb− Γbc a qx a . b Cl.r s= h r, hs ⎡⎣ ⎤⎦α hαl =(h r β∂ βhs µ−h s β∂ βhr µ) h µl C.r s l =0 Cl.r j=h r ϕ∂ ϕhj µ h µl C.r j l =0 ∂ ϕhi µ = 0 C.r s t Vm C.i j k Vn ∂ ϕhi µ =0 Γi j r = Γr j i ∂ϕhiµ =0 Γs j r = Γr j s Γr st Vm Γi j k Vn ∂ϕhiµ =0 2Γr s i =ηr tC.i s t −η s tC.i r t 2Γi r s =ηs tC.i r t +η r tC.i s t 2Γi j r =ηr sC.i js ∂ϕhiµ =0 (ha α∂ αhb β −h b α∂ αha β)=C .a b c h c β ha αh b β(−∂ αhβ d +∂ βhα d)=C .a b d Fα β d ≡(∂αhβd − ∂ βhαd)=−C.a b d h αahβb ∂τhϕr − ∂ϕhτr+hϕshτt C. t s r = 0 ∂νhϕr − ∂ϕhνr +hνshϕu C. s u r + hνihϕu C. i u r = 0 C. i u r = hi ν hu ϕ Frϕν +hiϕhuτ Frτ ϕ
! (1.18d) In order to get some simplifications we have used several times the condition : ! (1.19a) It will be assumed in all what follows. It implies :
! (1.19b) One will need the following results :
! (1.20) whatever the torsion is, and
! (1.21) where : ! is the determinant of the ! metric tensor.
Summary of this section.
As in Kaluza-Klein theories, the space-time is assumed to be an ! dimensional manifold ! , locally of the form : ! . ! is supposed to be invariant under the action of a group of motion ! . The basis vectors of the local orthonormal frames are supposed to satisfy (1.5) , (1.6) : ! .The consequences are given by the relations (1.16). We also impose the condition (1.19) : ! .
2. Spinors in !
, action of the !
invariance group.
In this chapter we study the action of the invariance group ! on the spinors of ! . In the following, the ! matrices are supposed to satisfy the constraints :
! and : !
A construction of the ! matrices in ! is presented in appendix A.
2.1 Spinor transformations.
Let : ! be a spinor field defined with respect to a family of orthonormal frames ! . If one performs an infinitesimal transformation (1.3) ! , ! remains the same in the transformed frames ! which we call adapted frames. In order to know how the spinors are transformed we have to expressed the adapted frames with respect to the local frame ! .
We name : ! the spinors defined with respect to the family ! and we set : ! and ! (2.1) C. i j s = hi νh j ρF ρν s +(hi νh j ϕ − h i ϕh j ν) F ϕν s +hi ωh j ϕ F ϕω s ∂ϕhi µ =0 ∂ϕhρj = 0 ∂αha α =− ha β∂ β g g − C. a e e ∂ϕhr ϕ =− hr ϕ∂ ϕ gVm gVm − C. r s s gVm V m n+m V =Vn+m Vn⊗ Vm Vm G hr µ = 0 ↔ hϕi =0 ∂ϕhi µ =0 Vn+m Vm G Vn+m γa γ0+=γ0 γ0γa+γ0 =γa γa Vn+m ψ (x) ha !"! (x)
{ }
f : x→ x' ψ(x) h 'a ! "! (x '){
}
ha !"! (x '){
}
ψ' ha !"! (x '){
}
ψ'= Λ−1ψ h 'a ! "! (x ')=A−1a b hb !"! (x ')The infinitesimal transformations (1.3) , with constant coefficients, gives, using (1.9b) : !
And, at first order : ! , ! (2.2) Usually the transformation of spinors is performed for transformations which conserve the orthonormality of the frames, such as space-time rotations. Here ! is a group of rigid transformations for ! , not necessarily for ! .
We shall require that the various tensors built with the spinors transform like tensors : For scalars : !
For vectors : ! which give the constraints : !
therefore : ! (2.3a) and : ! (2.3b) For infinitesimal transformations, we set : ! (2.4) and at first order : !
since : !
we then have : ! (2.5) The constraints (2.3a) and (2.3b) become respectively :
! (2.6a) and : ! (2.6b) What can be said about ! ?
Let us assume that one can write : ! , where ! is a product of ! Dirac matrices in which all the indices are different. If ! , ! is simply a complex number times the unit matrix. For each sequence with ! fixed , the commutator of (2.6b) projects on sequences of type ! , and since the right member of (2.6b) is of degree 1 : ! ou ! . Therefore : ! where : ! . When used back in (2.6b), one gets : ! , which implies : ! , and then ! should be a group of rigid motion for ! , which is not true.
Despite that, we consider : ! (2.7) even if ! is not a group of rigid motion, and calculate : ! .
Using the relation (1.10) one obtains : ! (2.8) which shows that ! satisfy the Lie algebra relations of ! , but we have to check that ! satisfies the constraints (2.6b) . We get : ! . In order to get rid of the difficulty explained above we shall restrict the indices ! in (2.7) to those
corresponding to the part of the local orthonormal frames tangent to ! , and we set : ! (2.9) h'a ! "! (x ')=A−1a b hb !"! (x ')=ha !"! (x ')−ηxqx a. b (x) hb !"! (x ') A−1a b =δ a b −ηx qx a . b Aa b =δ a b +ηx qx a . b G Vm Vn+m ψ'ψ'=ψ ψ ψ ' γaψ '= A−1b a ψ γbψ ψ+Λ−1+γ0Λ−1ψ = ψ+γ0ψ Λ+γ0Λ=γ0 Λ−1γbΛ= Aa bγa Λ= I+ηxMx Λ−1= I−ηxMx ψ '(x')=ψ '(x)+ηx Xx α∂ αψ '=Λ−1ψ (x) ψ '(x)=ψ (x) − ηx (Xx +Mx)ψ =ψ (x) − η x Txψ Mx +γ0 = −γ0 Mx γa , Mx ⎡⎣ ⎤⎦=qxb. aγb Mx Mx =αhγ h γh =γα1 ...γαh h h=0 γh h h± 1 h=0 h=2 Mx=S+αc dγ cγd c≠ d 2 (αa b −αb a)=qx b a qx a b = −qx b a G Vn+m T!x = Xx− 1 4 qx c d γ cγd = Xx+M"x G ⎡⎣T! , Tx !y⎤⎦ T! , Tx !y ⎡⎣ ⎤⎦ =C. x y z T!z T!x G M!x γa , M!x ⎡⎣ ⎤⎦ = 12 (qxb. a − qx.b a )γb c, d Vm T!x = Xx − 1 4 qx r sγ rγs = Xx +M"x
We can now reconsider the preceding calculations. Using : ! , one gets : !
Now, with (1.12c), one can use (1.10) to show that (2.8) is satisfied. Recall that, in order to obtain this result, we have used the hypothesis that ! is a rigid motion group of ! . It remains to check that the conditions (2.6a) an (2.6b) are satisfied. Condition (2.6a) is a direct consequence of (A11) et (A12). For the constraint (2.6b) one must check that : !
If ! , the commutator is zero, and : !
but from (1.12) : ! and ! if : ! . With this last condition, (2.6b) is satisfied for ! . If ! , the commutator is :
!
and the constraint is satisfied if : ! , which is true if ! is a group of rigid motion of ! .
Finally : ! satisfies the Lie algebra relations of ! and the constraints (2.6) if ! is a group of rigid motion of ! and if : ! (2.10) From (1.9b) and (1.12b) , this last condition means that ! is unchanged under the action of ! (! ) , but that ! change (! ) . This fact allows to build gauge fields in Kaluza-Klein theories by defining (1.8).
2.2 Hermitic conjugate of !
.Let us consider the scalar product : ! and let us look at the action of ! :
! (! )
!
For a group of rigid motion which preserves the volume elements, one obtains :
! (2.11) and therefore : ! is anti-hermitic.
2.3 Transformation of the covariant derivative and of the connexion.
We shall first check that the Dirac operator: ! qx r s+qx s r =0 Mx ! , M!y ⎡⎣ ⎤⎦= 1 4 (qx t r qy s . t − q y t r qx s . t )γrγs G Vm γa , M!x ⎡⎣ ⎤⎦ = γa ,−1 4 qx r sγ rγs ⎡ ⎣⎢ ⎤ ⎦⎥ = ? qxb. a γb a=i qxb. i γb = qxt . i γt + qx j. i γ j qxt . i = 0 qx j. i = 0 ∂ϕhi µ = 0 a=i a=t γt ,1 4 qx r sγ rγs ⎡ ⎣⎢ ⎤ ⎦⎥= 1 2(qx . s t − q x s. t )γ s qx r s = −qx s r G Vm T!x = Xx− 1 4 qx r s γ rγs = Xx+M"x G G Vm ∂ ϕhi µ =0 hi µ G qx j. i = 0 hi ϕ qxi. r ≠ 0 T!x S= ϕ ψ dV Vm
∫
T!x ϕ T! ψ gx =ϕ (Xx ϕ∂ ϕ − 1 4 qx r sγ rγs )ψ g r≠s =− (Xx ϕ∂ ϕ− 1 4 qx r sγ rγ s )ϕ ψ g+∂ϕ(ϕ Xx ϕ ψ g )−ϕ∂ ϕ(Xx ϕ g )ψ ϕT! ψ gx =− T! ϕ ψ gx +∂ϕ(ϕ Xx ϕ ψ g) T!x γa ha α(∂ α + Γα −Sα)ψtransforms as expected. Let us replace ! by ! , and use (2.3b) : !
!
where we have written : ! (2.12) setting : ! , one gets the expected transformation law :
! (2.13) Now, using (2.9), and at first order : !
we separate the connexion into 3 contributions : ! then : !
!
The first commutator is zero, therefore : ! (2.14a) The last two terms are of the form : ! for the group ! whose Lie algebra elements are represented by : ! , and therefore represent a gauge transformation in ! . (2.14b) At last : ! (2.14c)
2.4 Hermiticity of the Dirac operator.
We define : ! (2.15) and we consider the scalar product : ! (2.16) In order to justify this expression we recall that expressions of the type : ! , where all the indices are différents, are rank ! tensors with respect to local orthonormal frame rotations, and then : ! transforms like a scalar for these transformations. We have : !
!
the third term on the right is ( with ! ) : !
! Using the relation : !
ψ Λψ ' γa ha α(∂ α +Γα − Sα)Λψ '=γaha αΛ (∂ α+Λ−1ΓαΛ+Λ−1∂αΛ − Sα)ψ ' = ΛAb a γb ha α(∂ α + Γ'α−Sα)ψ' Γ'α =Λ−1Γ αΛ+Λ−1∂αΛ fb !"! =Ab a ha !"! γa ha α (∂α + Γα −Sα)ψ = Λ γb fb α (∂α + Γ'α−Sα)ψ' Γ'α =Γα+ηx Γ α, Mx
[
]
+∂α(ηxMx) Γα =Γi jα γiγ j 4 +Γi rα γiγr 2 +Γr sα γrγs 4 Γ'α =Γα−ηx qx r s Γi jα γiγ j 4 , γrγs 4 ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ − ηx qx r s Γi tα γiγt 2 , γrγs 4 ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ −ηx qx r s Γu vα γuγv 4 , γrγs 4 ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ − ∂α( ηxqx r s γrγs 4 ) Γ'i jα = Γi jα S−1ΓαS+S−1dS SO(m) γrγs 2 Vm Γ'i sαγ iγs = Γ i sαγ iγs −1 2η x qx t s Γi .α t γiγs D =γaha α (∂α + Γα −Sα)ψ S= ϕ ψ dV Vn+m∫
= ϕ+ γ0ψ dV Vn+m∫
ϕ γα1...γαhψ h ϕψ ϕ Dψ g=∂α(ha αϕγaψ g) − h a α∂ αϕ+γa+γ0ψ g +ha αϕ+γa+γ0 (Γcdα γcγd 4 − Sα)ψ g − ∂α(ha α g )ϕ γaψ c≠d ha αϕ+γa+ (Γcdα γc+γd+ 4 − Sα)γ 0ψ g = ha α (Γcdα γdγc 4 − Sα)γ aϕ ⎛ ⎝⎜ ⎞ ⎠⎟ + γ0ψ g = −ha αγa (Γcdα γcγd 4 +Sα)ϕ+Γ. d a a γdϕ ⎛ ⎝⎜ ⎞ ⎠⎟ + γ0ψ g Γ.a b b = 1 g∂α(ha α g )+2 S.β γ γ ha βwe finally obtain : ! (2.17)
2.5 The Dirac operator and the commutators.
We shall now write the Dirac operator (2.15) using the local orthonormal frame basis vector commutators. From (1.15d) one has :
!
which, we recall, is true for rigid frames . The torsion dependent terms are equal to: !
In this expression ! but ! is free. Separating the various cases, one gets :
! (2.18) where all the indices in the first term are different.
One can do the same thing with the commutation coefficients and finally obtain :
! (2.19)
where all the indices in the first two terms on the right are different.
Note that , from (1.16a) , ! , which means that commutators ! belong to the ! tangent space, and that, from (1.16d) , these commutators are calculated on ! only. Then (2.19) is true also on ! only :
! (2.20)
where : ! , and where : ! in the two first right member terms .
Summary.
Under the action of the group of rigid motions ! , the spinors are transformed according to : ! where : !
satisfies the ! Lie algebra relations and the conditions (2.6a) et (2.6b), if the constraint (1.19a) is satisfied. ! is anti-hermitic .The covariant derivative of a spinor transforms like a spinor, the Dirac operator (2.15) is anti-hermitic ( ! is hermitic).
3. The gravitational field as a gauge field.
Let us consider the Dirac matrices ! which represent the basis elements ! of the Clifford algebra : !
The commutators : ! represent the generators of the rotation group. They satisfy : ! (3.1a) ϕ Dψ g = −Dϕ ψ g + ∂α(ha αϕ γaψ g ) γa ha αΓ α =Γcd a γaγcγd 4 = 1 8(Ca cd − Cd a c− Ccd a)γ aγcγd + 1 4 (−Sa cd +Sd a c+Scd a)γ aγcγd 1 4 Sa cd (γ aγcγd + 6ηa dγc ) c≠d a 1 4 Sa cd γ aγcγd + S. cd d γc Γcd a γaγcγd 4 − Saγ a=−1 8Ca cdγ aγcγd + 1 4 Sa cd γ aγcγd −1 2C. cd d γc C.r s l = 0
[
hr, hs]
V m Vm Vm Γst r γrγsγt 4 −Srγ r= −1 8 Cr st γ rγsγt +1 4 Sr st γ rγsγt −1 2 C. r s s γr Sr=Ss. r s ≠Sc. r c r≠s≠t ≠r G ψ'(x)=ψ(x)−ηx T!xψ T!x = Xx− 1 4 qx r s γ rγs = Xx+M"x G T!x i D γa {ea} ea.eb +eb.ea =2ηab Rab = 1 4 γ a ,γb ⎡⎣ ⎤⎦ [Rab, Rcd]= −ηa dRcb−ηa cRb d −ηb dRa c−ηb cRd aand : ! (3.1b) ! (3.1c) This set of relations is a graded Lie algebra which satisfies Jacobi’s identities. The ! and ! matrices are the elements of a representation ! of an algebra defined by the relations (3.1) and named ! , where ! are the basis elements of this algebra. To this algebra we associate a gauge field:
! (3.2) where : ! and ! are differential forms of degree 1 :
! ! (3.3) and : ! , ! are arbitrary constants.
The meaning of these gauge fields is the following : the fields ! are the connexion coefficients defined with respect to a family of local orthonormal frames, an the 1-forms ! are the coordinates, in the neighborhood of a given point ! , with respect to these frames. This can be understood by computing the curvature 2-form (B.8) :
! (3.4) !
! !
and since the gauge fields (3.3) are 1-forms, one has :
!
With the relations (3.1) this becomes :
! (3.5)
We can set : ! (3.6)
Then : ! (3.7a)
With the above interpretation of the gauge fields, the first two terms in the first brackets correspond to the usual curvature 2-form ! , and the second brackets contains the structure equations : ! , where ! represents the torsion 2-form (B.6) : ! (3.7b) The standard minimum gauge field Lagrangian is :
! (3.8) where : ! is the Hodge’s star operator. Taking into account the ! matrices properties :
! , ! , ! (3.9)
where : ! is the spinor dimension, one has :
[γa,γb]=4 Rab [γa, Rcd]=ηa cγd −ηa dγc γa Rab Γ Γ(Xx) Xx W =α ωabR ab +βω aγ a ωab =−ωb a ωa ωab =ωabαdx α ω a =ηabω b =η abhα b dxα α β ωab ωa (x)=ηabωb(x) x G=dW +W ∧W G=(αdωabR ab +β dωaγ a )+(α ωcdR cd +βω cγ c )∧(α ωe f R e f +βω eγ e ) G=α dωabR ab +α 2 2 (ωcd∧ωe f R cd Re f +ω e f ∧ωcdR e f Rcd ) +βdωaγ a+αβ (ωcd∧ωeR cdγe+ω e∧ωcd∧γ e Rcd )+β2ω c∧ωeγ cγe G=αdωabR ab +α 2 2 ωcd ∧ωe f[R cd , Re f]+βdωaγ a+αβ ω cd∧ωe[R cd ,γe]+2β2ωa∧ωbR ab G=[αdωab +2α 2ω a f ∧ω.b f + 2β2ω a ∧ωb] R ab +β [dωa +2α ωa . e∧ω e]γ a α = 1 2 G= 1 2[dωab +ωa f ∧ω.b f + 4β2ω a∧ωb] R ab +β [dω a+ωa . e∧ω e]γ a Ω. b a = dω. b a +ω . c a ∧ω . b c dωa +ω. b a ∧ωb =Σa Σa G= 1 2[Ωab +4β 2ω a∧ωb] R ab +β Σ aγ a L=Tr(G∧ ∗G) ∗ γa Tr(RabRcd)= N 4 (η a dηb c−ηa cηb d ) Tr(Rabγc)=0 Tr(γaγb)= Nηab N
! (3.10a) !
The first term correspond to the Einstein-Hilbert Lagrangian of General Relativity. The third term represents the contribution of a cosmological constant, since this term is proportional to the volume element. The second term is quadratic and has the form of standard gauge field Lagrangian. The equation (3.10a) can also be re-written :
! (3.10b) !
where : ! is a Lagrange multiplicator.
In (3.10) the torsion is introduced naturally, not as an extra field, this is a direct consequence of the definition (3.2) where ! and ! are independent fields.
Until now, we have discussed the interpretation of the gauge fields introduced in (2.2), but it remains to see how these fields transform. Taking into account the algebra (3.1) , one
considers the infinitesimal transformations :
! where : ! et : ! .
The transformation law of gauge filed is : ! which gives : ! and makes the Lagrangian (3.8) invariant. With, at first order : !
one has, still at first order : !
!
and with the algebra (3.1) : !
and with : ! : !
If : ! , ! transforms like a vector under an (infinitesimal) rotation whose coefficients are the ! . In that case : ! and the field ! transforms like a gauge field with respect to rotations. The restriction of the gauge transformation makes ! transform like a vector. The 1-forms : ! are directly related to the basis vectors of the local frames. In the above description ! et ! are independent gauge fields. How does that modifies the Lagrangian of matter fields ? We shall suppose that ordinary matter fields are spinor fields, and therefore we shall consider the covariant derivative of such fields.
Let ! be a spinor field. The covariant derivative of a spinor field with gauge field is :
! L / N =−β2 Ωab ∧ ∗(ωa ∧ωb)− 1 8 Ω ab∧ ∗Ω ab −2β4 (ωa ∧ωb )∧ ∗(ωa∧ωb)+β 2 η a b(dω a +ω . c a ∧ωc )∧ ∗(dωb+ω . d b ∧ωd ) L / N = −β2 Ωab ∧ ∗(ωa ∧ωb)− 1 8 Ω ab∧ ∗Ω ab −2β4 (ωa ∧ωb )∧ ∗(ωa∧ωb)+β 2 Σa ∧ ∗Σ a+µ(dω a +ω . b a ∧ωb − Σa ) µ ωa ωab S=I+iεaγ a+ε abR ab εab =−εb a εa, εab ≪ 1 W '=S−1W S+S−1dS G '=S−1G S S−1= I− iεaγ a−ε abR ab W '=W +iα ωabεe R ab ,γe ⎡⎣ ⎤⎦+α ωabεef R ab , Re f ⎡⎣ ⎤⎦+iβωaεe γ a ,γe ⎡⎣ ⎤⎦ +β ωaεef γ a , Re f ⎡⎣ ⎤⎦+i dεeγ e+ dεef R e f ω 'a =ωa+εeaω e−ε a eω e+ i β(dεa+α ωa. eε e−α ω. a e ε e) α =1 / 2 ω 'a =ωa+εeaω e −ε a eω e+ i βDεa Dεa =0 ω a εe f εa =0 ωab ωa ωa = hαa dxα ωa ωab ψ Dψ =dψ +1 4ωcdγ cγdψ +βω aγ aψ → D αψ =∂αψ + 14ωcdαγ cγdψ +β h αbηb cγ cψ
The last term is not present in the usual covariant derivative of a spinor field. The Lagrangian of such a field is : !
where : ! means : Hermitic conjugate. If one uses the above covariant derivative, the contribution of the unwanted terms is (if ! is real) :
!
Therefore, the gauge field (3.2) gives the usual spinor field Lagrangian built with the usual covariant derivative : !
Remark : the above Lagrangian of a spinor field is built using the generator ! of the algebra (3.1). Why not use the generators of type ! instead ? A possible Lagrangian could be : ! , which is , taking into account the fact that : ! , ! , !
which gives the second order Dirac equation. In conclusion, the gauge field (2.2) does not introduces unwanted terms.
Summary.
The gauge field (3.2) associated to the algebra (3.1) leads to a gravitational Lagrangian which is more general than the Einstein-Hilbert one, introducing naturally a quadratic term and a cosmological constant. The 1-form fields ! and ! are independent of each other, and as a consequence, torsion may exist as an independent field.
4. The Dirac operator in !
.
4.1 The classical Dirac equation with gauge fields : from !
to !
.
The evolution of a spinor field in ! will be assumed to be governed by the standard « minimum » Lagrangian :
! (4.1) where : ! means : hermitic conjugate.
The construction of a possible representation of the Dirac matrices ! is detailed in appendix A . In all what follows ! is assumed to be even.
After integration of the Action on ! , one would like to recover the « macroscopic » Dirac equation with gauge fields and gravitational field. In other words, one would like to write : ! (4.2) Note that, when we write this equation in this way, we assume that the spinor ! is a « multiplet » of : ! spinors ! of ! of the form :
L=ψ ha αγa i Dαψ +h.c. h.c. β iβψ ha αγa hαbη b cγ cψ + h.c.=iβψ γaη a cγ cψ + h.c.∼i βψ ψ +h.c.=0 Dαψ =∂αψ +1 4ωcdαγ cγdψ γa Rab L=Daψ R ab Dbψ +h.c. R ab = 1 2γ aγb a≠b L= Dψ . Dψ −Daψ η ab Dbψ ωa ω a b Vn+m Vn+m Vn Vn+m L=ψ γaha α i Dαψ +(h.c.)+mass terms h.c. γa m Vm γi hi µ(∂ µ+Γj kµ γ jγk 4 − Sµ +Wµ)ψ =mψ ψ qm =2 m 2 ψ i V n
! (4.3)
In the Dirac equation (4.2), the gauge fields acts on the multiplets and not on the components of each ! . In other words, in equation (4.2) , the ! are supposed to be of the form :
! , where we have written : ! to mean it is the Dirac matrice number ! defined in a space-time of dimension ! .
The appendix A shows that is is possible to « diagonalise » the Dirac matrices of type ! , ! , in the form : ! (relations (A.23) , (A.24) , (A.27) , (A.17) ) .
The commutation relations : ! et : ! show that the : ! are good candidates to represent the ! Lie algebra generators. At last, note that, from now on, we shall suppose that the conditions (2.10) are satisfied.
4.2 Decomposition of the Dirac operator.
From (1.15) , one has : !
Using (2.18), the Dirac operator (2.15) is written : !
! (4.4) !
where, in the last term : ! (only the totally anti-symmetric part of the torsion tensor is taken into account).
Grouping the terms 3 et 7 , (4.4) is : !
! (4.5) !
where we have defined : ! (4.6) ! is Dirac operator on ! without torsion, but it is not the Dirac equation on ! because it does not include the time coordinate . This operator will be studied in chapter 6 when the mass term will be discussed .
The first term of the second line and the first term of the third line give, using (1.16f) : ψ = ψ1 ψ2 ! ψqm ⎛ ⎝ ⎜ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟ ⎟ ψi γ i Iqm ⊗γn i γ p a a p γn+m i 0≤i<n Iqm ⊗γn i γi ,γrγs ⎡⎣ ⎤⎦ =0 γiγ j ,γrγs ⎡⎣ ⎤⎦ =0 γ rγs 2 SO(m) Γa bc = Γa bc+Sa bc+Sbc a+Scba D =γa ha αD αψ =γihi µ(∂ µ +Γj kµγ jγk 4 +Γj rµ γ jγr 2 +Γr sµ γrγs 4 )ψ + γi hi ϕ(∂ ϕ +Γj kϕ γ jγk 4 +Γj rϕ γ jγr 2 +Γr sϕ γrγs 4 )ψ + γr hr ϕ (∂ϕ + Γstϕ γ sγt 4 + Γj kϕ γ jγk 4 + Γj sϕ γ jγs 2 )ψ +Sab c γaγbγc 4 a≠b≠c≠a D =γi hi µ(∂ µ+Γj kµγ jγk 4 +Γr sµ γrγs 4 )ψ + Dmψ +Γj r iγ iγ jγr 2 ψ +γ i hi ϕ(∂ ϕ +Γj kϕγ jγk 4 +Γr sϕ γrγs 4 )ψ +(Γj k rγ rγ jγk 4 + Γj s r γrγ jγs 2 )ψ +Sab c γaγbγc 4 Dm=γ r hr ϕ (∂ϕ + Γstϕ γ sγt 4 ) Dm V m Vm
!
!
where : ! in the first term on the right . Finally , using (1.16f) again :
!
for orthonormal frames, one has with (1.16f) : ! . Then :
! (4.7) The term before the last in (4.5) is :
!
then , by permutation of the Dirac matrices in the second term, and with (1.16f) :
! (4.8) The equation (4.5) becomes :
! (4.9) !
!
where : ! .
In order to assess the relative importance of the terms in (4.9), we shall write them as functions of the fields : ! .
From (1.16k), and with condition (1.19a), we have : ! With (1.7) , (1.18d) is :
!
we have also : ! (4.10) therefore : ! (4.11) In the same way (1.18c) is :
! (4.12) and (1.16i) : ! (4.13) Γj r iγ iγ jγr 2 + Γj k r γrγ jγk 4 = Γj r i γiγ jγr 2 + Γr k j γrγ jγk 4 = Γk r j + 1 2 Γr k j ⎛ ⎝⎜ ⎞⎠⎟γ rγ jγk 2 +Γ. r k k γ r 2 j≠k Γj r iγ iγ jγr 2 + Γj k r γrγ jγk 4 = − Γj k r γrγ jγk 4 + Γ. r k k γr 2 Γk r k = −Γr k k = −Γk k r =0 Γj r iγ iγ jγr 2 + Γj k r γrγ jγk 4 = − Γj k r γrγ jγk 4 Γj s rγ rγ jγs 2 = − Γs j r γrγ jγs 2 = − 1 2 Γs j r γrγ jγs 2 + Γr j s γsγ jγr 2 ⎛ ⎝⎜ ⎞ ⎠⎟ Γj s rγ rγ jγs 2 = Γ. j r r γ j 2 D =γi hi µ(∂ µ + Γj kµγ jγk 4 + Γr sµ γrγs 4 )ψ +Dmψ +γi hi ϕ (∂ϕ + Γj kϕ γ jγk 4 + Γr sϕ γrγs 4 )ψ −(Γj k r γrγ jγk 4 − Γ. j r r γ j 2 )ψ +Sab c γaγbγc 4 Γj k r=hrϕ Γj kϕ hµr 2Γi j r =ηr sCs. i j C. i j s = hi νh j ρ F ρν s +hνt ht ϕ F ϕ ρ s −hρt ht ϕ F ϕν s +hνr hr ωh ρ t ht ϕ) F ϕω s
{
}
C.r s t =h r ϕh s τ F τ ϕ t C. i j s = hi ν hj ρ Fρνs +hνrhρtC.r ts +hνthtϕ Fϕ ρs −hρt htϕ Fϕνs{
}
C. i u r = hi νh u ϕ F ϕν r − hi νh νtht ϕh u τ F τ ϕ r =hi ν(h u ϕ F ϕν r − hνt C.t u r ) 2Γr s i =ηr tC.i s t −η s tC.i r t =h i ν (η r ths ϕ −η s thr ϕ) F ϕν t −hνu(C r u s−Cs u r) ⎛ ⎝⎜ ⎞⎠⎟! (4.14) With (1.15g) one obtains :
! (4.15)
The basic hypothesis was that the space-time manifold was locally of the form ! where : ! is a compact space invariant by the action of a group of motion ! . We shall set : ! (4.16) where the fields : ! are the components of the differential forms associated to the orthonormal local frames for a manifold ! of unit curvature radius , and where : ! represents the curvature radius of ! at the point ! of ! . With (1.7) one then has : ! (4.17a) or , like in (1.7) : ! (4.17b) We now gather a few results useful for the coming calculations :
! (4.18) ! (4.19) ! (4.20) this term, which is the ninth of (4.9) contributes, as well as the second of (4.9) to terms of the form : ! . ! (4.21) With (4.16) : ! (4.22) and : ! (4.23a) ! (4.23b) ! (4.23c) ! (4.23d) from (4.12) and (4.18) : ! (4.23e) from (4.14) and (4.18) : ! (4.23f) The seventh term of (4.9) is ! , with : !
then : ! 2Γi r s =ηr tC.i s t +η s tC.i r t =h i ν (η r ths ϕ +η s thr ϕ) F ϕν t − hνu(C r u s+Cs u r) ⎛ ⎝⎜ ⎞⎠⎟ 2Γr sµ =2Γr s ihµi +2Γr s thµt =(ηr ths ϕ −η s thr ϕ) F ϕµ t +hµt C t r s V =Vn⊗ Vm Vm G hϕr =q(xµ) hϕr(xτ) hϕr Vm q(xµ) Vm
{ }
xµ Vn hϕr hs ϕ =q(xµ) h ϕ r hs ϕ =δ s r → hs ϕ =1 q hs ϕ (xτ) hϕ r hs ϕ =δs r , hϕ r hr τ =δϕτ C. t u s = 1 q (ht ϕ ∂ϕhu τ − hu ϕ ∂ϕht τ )hτs ≡ 1 qC. t u s 2Γr sµ = 1 q (ηr ths ϕ −ηs thr ϕ )∂ϕhµt +h µ tC t r s ⎛ ⎝⎜ ⎞⎠⎟ hi ϕ Γ j kϕ=hiϕhϕr Γj k r =−(hiµhtµhtϕ)hϕr Γj k r =−hiµ hµr Γj k r γiγ jγk 2Γj k r =ηr s hνj hkρ Fρν s +hνt hρu C.t u s +1 q ht ϕ (hνt∂ ϕhρs − hρt ∂ϕhνs)+ 1 q(qνhρ s − q ρhνs) ⎧ ⎨ ⎩ ⎫ ⎬ ⎭ hνr ∼ O(q) Γr sµ ∼ O(1) Γj k r ∼ O(q) hi ϕ Γ j kϕ∼ O(q2) hk ϕ =−h k µ hµrhr ϕ → h k ϕ ∼ O(1) Cr i s ∼ O(1) Γi r s ∼ O(1) ∼ hi ϕ Γ r sϕ Γr sϕ=hϕa Γr s a=hϕt Γr st =q hϕ t Γr st hi ϕ Γ r sϕ= −hiµ hµt htϕ q hϕ u Γr su = −hiµ hµt Γr stwith : ! (4.24) therefore: ! , and this term can be grouped with the third of (4.9) to give : ! . With (4.13) :
! (4.25)
The « macroscopic" Dirac operator (4.9) is therefore : !
! (4.26) where : ! in the first line, and ! in the last term.
Taking into account (1.7) , one can put together the last term of the first line and the first of the second line : !
We define, using , (4.18) : ! (4.27) The operator ! has the same properties as ! , and satisfies the relations (2.8) , if the ! are associated to rigid motions. With (1.11) this condition is :
! (4.28) In order to consider the operators ! as the operators of the Lie algebra of a group ! , ! must be the equivalent of the parameter space of this group. ! is an invariance group of ! , it is sufficient that ! is the quotient group of ! by the (invariant) stability group of ! .
Note : if the Lie algebra of ! is semi simple and compact, the Killing form of the real Lie algebra is negative definite by definition, and the structure constants are totally anti-symmetric. In the following we shall assume that ! is a simple group.
From (4.22) we define : ! (4.29a) then , using (4.23d) : ! (4.29b) This hypothesis is more constraining than (4.22), it means that ! does not depend on the internal coordinates of ! .
Then : !
represents the field ! contribution. This field transforms like ! in section 1.1.
Now we consider the second term of (4.26) : ! 2Γr st=Ct r s −Cst r−Cr st = 1 q
(
Ct r s −Cst r −Cr st)
hi ϕ Γ r sϕ∼ O(1) Γr siγ iγrγs 4 2Γr s iγ iγrγs =γih i ν (η r ths ϕ −η s thr ϕ) F ϕν t − hνu(C r u s− Cs u r) ⎛ ⎝⎜ ⎞⎠⎟ γrγs D =γihi µ ∂ µ +Γj kµγ jγk 4 + 1 2(ηr ths ϕ −η s thr ϕ ) Ftϕµ − hµt Cr t s ⎛ ⎝⎜ ⎞⎠⎟γ rγs 4 ⎛ ⎝⎜ ⎞ ⎠⎟ψ +γi hi ϕ∂ ϕψ +Γ. j r r γ j 2 ψ +Dmψ +Sab c γaγbγc 4 r≠s a≠b≠c≠a γi hi ϕ∂ ϕ −γihi µ hµt Cr t s γrγs 4 =−γ i hi µ hµt ( ht ϕ∂ ϕ +Cr t s γrγs 4 ) T!t = ht ϕ ∂ϕ +Cr t sγ rγs 4 T!t T!x ht ϕ ∂ϕ Cr t s+Cs t r =0 T!t Q Vm G Vm Q G Vm G G hνr =−q(xµ ) Aνr (xρ) hk ϕ =h k µA µrhr ϕ =Ak r hr ϕ hνr Vm γi hi ϕ∂ ϕ −γihi µ hµt Cr t s γrγs 4 =γ i hi µ AµtT!t Aνr (xρ) Wµx Γj kµ =hµi Γj k i+hµr Γj k r =hµi Γj k i +O(q2)from (1.16h) and (1.19a) , ! and ! do not depend on the ! , then, at the macroscopic level : ! (4.30) With the hypothesis (4.29), equation (4.26) becomes :
!
Now, from (1.15f) and (4.28) : ! , and it remains, at the « macroscopic »
level : ! (4.31)
where : ! in the last term.
In order to obtain the Dirac equation (4.2) from (4.31) , one must « diagonalise » the first term of (4.31). This can be done by multiplying (4.31) by the operator (A24).
Summary.
In order to put the Dirac operator ! in the form (4.2) we have done 2 hypothesis :
the hypothesis (4.29) : ! , and that ! is equivalent to the parameter space of ! , with the contraints (4.28) . With this condition on ! , the operators ! can be
considered as belonging to a representation of the Lie algebra of ! , moreover they commute with the « spatial » part of the Dirac operator (4.31).
5 . Gauge field Lagrangian.
5.1 Curvature tensor decomposition.
The curvature 2-form is : !
!
symmetrizing and using (1.15a) :
! (5.1)
which is valid with or without torsion. With the definitions (1.15) the connexion is re-written : ! (5.2) where : ! is the contorsion tensor and : ! is the part of the connexion depending on the Christoffel symbols. The curvature tensor is then :
! ! ! we set : Γj k i hiµ
{ }
xν γi hi µ (∂µ +Γj kµγ jγk 4 ) , T!t ⎡ ⎣ ⎢ ⎤ ⎦ ⎥=0 D =γi hi µ ∂ µ +Γj kµγ jγk 4 +Aµ t T!t ⎛ ⎝⎜ ⎞ ⎠⎟ψ +Γ. j r r γ j 2 ψ + Dmψ +Sab c γaγbγc 4 Γ. j r r = −Cr. j r =0 D =γihi µ ∂ µ+ Γj kµγ jγk 4 +Aµ t T!t ⎛ ⎝⎜ ⎞ ⎠⎟ψ + Dmψ +Sab c γaγbγc 4 a≠b≠c≠a D hνr = −q(xµ) Aνr(xρ) Vm G Vm T!t G Ωa b =d(Γa b gω g )+Γa e fω f ∧ Γ . b g e ωg =dΓa b g∧ω g+Γ a b g(Σ. c d g − Γ . d c g )ωc∧ωd+Γa e fΓ. b g e ωf ∧ωg 2Ωa b = hf α∂ αΓa b g− hg α∂ αΓa b f +Γa e fΓ. b g e − Γ a e gΓ. b f e − Γ a b eC. f g e(
)
ωf ∧ωg Γa b c = Γa b c+Sa b c Sa bc Γa b c 2Ωa b = hf α∂ αΓa b g−hgα∂αΓa b f + Γa e fΓ. b g e − Γa e gΓ. b f e − Γa b eC. f ge(
)
ωf ∧ωg + hf α∂ αSa b g− hgα∂αSa b f +Γa e fS. b g e +Sa e fΓ. b g e − Γa e gS. b f e − Sa e gΓ. b f e(
)
ωf ∧ωg − Sa b eC. f ge + Sa e fS. b g e − Sa e gS. b f e(
)
ωf ∧ωg! (5.3) then , with (1.15h) :
!
! (5.4) which separates the torsion terms.
In the following we shall calculate the part of the curvature tensor corresponding to the first line of (5.4), or equivalently, we shall suppose for a while, that the the torsion is null. We now write separately the curvature tensor components.
! ! (5.5) ! ! ! (5.6) ! ! ! (5.7) !
5.2 Computation of the components.
In order to compute the components of the curvature tensor we shall need the following elements :
!
!
with : ! , and (4.29) , one has :
! ! ! and finally : ! (5.8) where : ! (5.9) With (1.16k) : ! (5.10a) which satisfies : ! (5.10b) DfSa b g =hαf ∂αSa b g− Γ. a f e Se b g − Γ. b f e Sa e g− Γ. g f e Sa b e 2Ωa b = hf α∂ αΓa b g−hgα∂αΓa b f + Γa e fΓ. b g e − Γa e gΓ. b f e − Γa b eCe. f g
